Advances in Applied Probability

On the total length of external branches for beta-coalescents

Jean-Stéphane Dhersin and Linglong Yuan

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Abstract

In this paper we consider the beta(2 - α, α)-coalescents with 1 < α < 2 and study the moments of external branches, in particular, the total external branch length Lext(n) of an initial sample of n individuals. For this class of coalescents, it has been proved that nα-1T(n)D T, where T(n) is the length of an external branch chosen at random and T is a known nonnegative random variable. For beta(2 - α, α)-coalescents with 1 < α < 2, we obtain limn→+∞n3α-5 E{(Lext(n) - n2-αE{T})2} = ((α - 1)Γ(α + 1))2Γ(4 - α) / ((3 - α)Γ(4 - 2α)).

Article information

Source
Adv. in Appl. Probab., Volume 47, Number 3 (2015), 693-714.

Dates
First available in Project Euclid: 8 October 2015

Permanent link to this document
https://projecteuclid.org/euclid.aap/1444308878

Digital Object Identifier
doi:10.1239/aap/1444308878

Mathematical Reviews number (MathSciNet)
MR3406604

Zentralblatt MATH identifier
1328.60195

Subjects
Primary: 60J28: Applications of continuous-time Markov processes on discrete state spaces 92D25: Population dynamics (general)
Secondary: 60J25: Continuous-time Markov processes on general state spaces 60J85: Applications of branching processes [See also 92Dxx]

Keywords
Coalescent process beta-coalescent total external branch length Fu and Li's statistical test

Citation

Dhersin, Jean-Stéphane; Yuan, Linglong. On the total length of external branches for beta-coalescents. Adv. in Appl. Probab. 47 (2015), no. 3, 693--714. doi:10.1239/aap/1444308878. https://projecteuclid.org/euclid.aap/1444308878


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